Pregnancies don't pause when you play another household.
But if you want and can use mods, there is a setting in mccc that let's you choose the name of sims born in a household you are not currently playing. I have this option on and get a pop up every time a sim gives birth so I can choose the name.
Side note, I as off building a house for a sim family and in my absence they had their baby! Now I can't move them in. Can you not move households while there are babies in the house?
Are you sure about that? How are you trying to move them? I think I was able to evict a household with a baby once from the neighbourhood screen and then move the household to a different lot. But it is quite a while ago so maybe the update changed it.